Talk:Cross-cap

It seems that most of the problems below have been dealt with already, so I'm removing the "disputed accuracy" banner on the main page. Feel free to restore it, fix the problems, etc. ScottMorrison (talk) 04:37, 6 November 2011 (UTC)

The last sentence of the first paragraph is false. How should I fix this problem? sam Mon Aug 16 16:58:40 CDT 2004

This is referred to in Möbius strip: if you understand what's written there, try copying it over here :-) --Phil | Talk 08:01, Nov 4, 2004 (UTC)

Accuracy of the article

There is virtually nothing in the article (other than the lovely graphics) that is not false. Starting with the misguided attempt to roughly define a cross-cap in the first sentence.

• First of all, a cross-cap is never topologically equivalent to a Moebius strip. It is a continuous image by a certain type of map of the (closed) Moebius strip into 3-space, a map that has an open interval's worth of double-points.

• As a topological subspace of 3-space, it is the space obtained by starting with a closed disk D², choosing an interval in the disk's interior -- say the image of [-1,1] via an embedding h: [-1,1] → int(D²) -- and then identifying the points h(x) and h(-x) with each other for all x in (0,1].

• A cross-cap may have a boundary that is a round (perfect) circle, but is required only to have a boundary that is an unknotted simple closed curve.

• Further, there are continuous deformations of the usual picture of a Moebius band to a picture that is still topologically a Moebius band embedded in 3-space (i.e., with no self-intersections), such that its boundary is a perfect circle. So this perfect-circle-boundary property in no way characterizes a cross-cap.

• The word cross-cap has been erroneously used to mean a cross-cap with a disk glued on to its boundary (making a continuous image of a projective plane). But this is an error, and it should not be perpetuated in this article. Rather, the article should warn people to avoid this misuse of the word.Daqu 00:53, 4 December 2006 (UTC)Daqu 01:18, 4 December 2006 (UTC)

Please correct this. Many people are being confused and misinformed. I am researching cross-caps, and a preliminary look at this page left me confused.--Quarkrider (talk) 21:50, 21 June 2011 (UTC)

Better late than never, I've now corrected many errors in the first paragraph. No further errors caught my attention at this time.Daqu (talk) 06:19, 19 January 2013 (UTC)

Parametric Equations

The stated parametric equations do not give the shown image. The fraction 2/3 should be removed in the formula for Z[u,v]. —Preceding unsigned comment added by 81.210.243.30 (talk) 14:05, 24 June 2008 (UTC)

Yes. 91.78.182.60 (talk) 15:19, 19 August 2010 (UTC)

Cross-Cap not (necessarily) an Immersion

Immersion means that something is locally an embedding. The pictures shown fail to be an immersion at the singular points (the image of any small neighborhood around these points in the domain will self-intersect). The cross-caps I usually saw in math books never were immersions. Furthermore, the picture shows a cross-cap (which is a map from the disk to the 3-sphere) with a disk glued on, so the label is misleading. 99.38.248.12 (talk) 23:59, 5 August 2010 (UTC)

What is the projective plane doing here?

Now that the nature of the cross-capped disk has been exposed as a model of the projective plane, I think that section no longer belongs in this article.

For example the same surface may be obtained by joining the edge of a Möbius strip to itself. We all know that the projective plane may be divided into a Möbius strip and a disc. If we shrink the Möbius strip to a line, we get the model shown of a self-intersecting disc. But we can alternatively shrink the disc to a point (via the Jordan curve theorem), which is topologically equivalent to zipping the edge of the Möbius strip to itself.

All this has precious little to do with cross-caps, but much to do with the topology of the projective plane.

— Cheers, Steelpillow (Talk) 09:33, 21 April 2012 (UTC)

What else to say?

So, what should this article discuss instead? Some suggestions:

On any non-orientable surface in R^3, a pair of cross-caps may be transformed into a handle, and vice versa. The Klein bottle is, or used to be, a minor exception, I'm not sure if it still is?

The cross-cap is a good example of a surface which is neither an embedding nor an immersion. I don't know if there is an appropriate term to use, beyond very general words like "injection" or "mapping".

While the cross-cap and Möbius strip are not homeomorphic, they have the same intrinsic topology (e.g. Euler characteristic and number of boundaries). Can the correct terminology for this be added?

[edit] although the Whitney umbrella is linked to, its relationship to the cross-cap could be made explicit.

— Cheers, Steelpillow (Talk) 09:44, 21 April 2012 (UTC)

Many issues

Hi, I don't do that usually, but I feel like I need to participate here. This article has many serious issues, it's one of the worst ones I've read in mathematics. The only good thing about it is the nice pictures. The most serious issue is that it is incomprehensible and confusing. It took me way too much time to make sense out of it. The intro says it is a Möbius strip with a self-intersection, but all the pictures show the "closed version" of it, which are alternatively called "crossed-capped disk", "cross-cap", or "closed cross-capped disk". Not a single picture of the Möbius strip version. Most sentences are inaccurate, confusing or absurd. Let's have a look at each one, taking them one a time:

• "In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection".

This is inaccurate (what is a model of a surface) and poorly said. But it is not horribly bad yet.

• "This self intersection precludes the cross-cap from being topologically equivalent (i.e., homeomorphic) to a Möbius strip."

This is stupid, sorry to say, according to a precise definition of "topologically equivalent", this statement is either false or without the least interest.

• "The term ‘cross-cap’, however, often implies that the surface has been deformed so that its boundary is an ordinary circle."

What?? (and "however", relating to?) What's the point here?

• "The term cross-cap is also inaccurately used to refer to the closed surface obtained by gluing a disk to a cross-cap. This is, in fact, a cross-cap glued to a disk."

So, if I follow the logic, let's only consider the "inaccurate" version now? And, seriously, am I the only one to see a slight redundancy between these two sentences?

• "A cross-cap that has been closed up by gluing a disc to its boundary is an immersion of the real projective plane."

No, it is not! How dare you? ;) As was already pointed out by other people here, it is not immersed. And not only this statement is false and misleading, but there is not one mention in the whole article that there is one (or more) singular points, which I guess is actually very relevant.

Okay, I think I'll stop here! — Preceding unsigned comment added by 78.227.79.96 (talk) 21:14, 19 December 2012 (UTC)

OK, so I moved the stuff about the real projective plane to where it might at least belong. There's not much left behind to criticise. — Cheers, Steelpillow (Talk) 21:40, 20 December 2012 (UTC)

Actually, there is, the critics I wrote above are still valid, I believe. If I may, here's a personal attempt to improve it (but it definitely needs some more work), something like:

In mathematics, a cross-cap is a (slightly ambiguous) term refering to a surface which is topologically a real projective plane, sometimes with a disk removed (in that case, it is topologically a Möbius strip). More precisely, a cross-cap may either refer to:

• A topological surface that is homeomorphic to the real projective plane (sometimes with a disk removed), as in the following statement: "any compact surface without boundary is homeomorphic to a sphere with some number of handles and at most two cross-caps glued on it" (this is the classification theorem for surfaces).
• a self-intersecting mapping from the real projective plane (sometimes with a disk removed) into three-dimensional space. (see pictures below, or see other article)

Then I think you could move back or copy part of (or all of) what there was here before (or not). Anyway, there definitely needs to be links between the two articles. Or, this article could be deleted (I'm not particularly in favor of that, on the other hand the article "real projective plane" is already an article about the topological projective plane, as opposed to the article "projective plane").2A01:E34:EE34:F600:646C:7E97:2184:3D13 (talk) 01:39, 21 December 2012 (UTC)

Wow, that's the first IPv6 address I've seen "in the wild"! So much wrong with such a short article. I have never seen the closed surface "cross-cap plus disk" referred to as a plain "cross-cap" - it is well known to be an example of the real projective plane and the claim otherwise is unreferenced, so I suspect the ambiguity is misplaced - it's just a plain mistake by the original editor. Or, am I ignorant and the claim is perfectly respectable and needs reinstating and referencing?

Meanwhile I have had a hack at the current version (including removal of the unreferenced claim). As an amateur, I always get confused over precise topological definitions - homeomorphism, homology, embedding, immersion, etc. etc. and the meaning of the phrase "topologically equivalent" sometimes seems as plastic as the rubber sheets themselves. Every definition of some term always invokes some other equally obscure term and/or some equations that mean nothing if you have not been on the right foundation course. So I am not qualified to say if the odd error of fact or contradiction still lurks in the article. — Cheers, Steelpillow (Talk) 11:03, 21 December 2012 (UTC)

OK, let me take issue with this:

An important theorem of topology, the classification theorem for surfaces, states that each two-dimensional compact manifold without boundary is homeomorphic to a sphere with some number (possibly 0) of "handles" and 0, 1, or 2 cross-caps.

If a cross-cap is a singular surface in 3-space, this makes no sense. The surface must have 0, 1, or two Mobius bands glued in (though there is probably a better way to say it).--96.237.56.211 (talk) 01:08, 27 April 2021 (UTC)

The way I understand it, each 2 manifold is homeomorphic to either an orientable 2 manifold which is a shere with zero or more handles (zero corresponding to a sphere properly speaking and p=1 to a torus), or else a nonorientable manifold, which is a sphere with one or more circular regions removed and as many moebius strips attached to the respective boundaries of these regions. The case q=1 is the projective plane. At least in my topology class, attaching a crosscap was synonymous with the attachment of a moebius strip to such a boundary — Preceding unsigned comment added by 2A01:CB0C:CD:D800:ECF7:4FE2:6C28:CA9F (talk) 13:05, 3 August 2022 (UTC)

If there is only one illustration accompany this article, it should not be of something else!

There is only one illustration with the article, and it is inexplicably not of a crosscap. Instead it is the image of a mapping of the projective plane into R³ that includes a crosscap.

A much better idea would be to include a picture of a crosscap, not of something else.65.157.43.50 (talk) 00:56, 15 August 2020 (UTC)